The Golden Mean & The Fibonacci Sequence

 

Fibonacci numbers are also important in art.  The ratio between successive Fibonacci numbers approximates an important constant called “the golden mean” or sometimes phi, which is approximately 1.61803. The higher you go in the Fibonacci sequence, the more closely the ratio between two successive numbers in the sequence approximates phi. (By the way phi2=phi + 1!)

the golden ratio = 1.61803399

Animated GIF of successive rectangles built on squares with sides that are Fibonacci numbers sprialling outwards

 

A rectangle whose sides are in the proportion 1 : 1.61803 is supposed to be the most aesthetically perfect rectangle (the “golden rectangle”). The Parthenon in Athens has such a rectangle as its face, and phi is said to have figured in the construction of the Great Pyramids. The “golden section,” in which a line is divided into segments of lengths in the ratio 1 : .61803 is supposed to be an aesthetically ideal way to divide a line.

Numerous artists have used the golden section in their works, as well as composers, including (perhaps) Beethoven and Mozart.

Nautilus shell, cross-sectionNautilus-like spiral inscribed over spiral of Fibonacci squares

The golden section is a line segment divided according to the golden ratio: The total length a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b.

 

Dividing a line segment according to the golden ratio

 

 File:History of Gold. Rersum 2007.jpg


 

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